Mathematical Truth

As Zach notes below, the topic for the meeting on Monday is the ontology and epistemology of mathematics, and the reading is an influential paper by the Princeton philosopher of mathematics Paul Benacerraf.

Here's some background that will help in understanding Benacerraf's argument. First, the ontology of mathematics is concerned with the nature of the objects that mathematicians study. For example, what are numbers? Do they exist independent of the human mind or are they creations of the mind? The epistemology of mathematics concerns how we come to have mathematical knowledge. Benacerraf's main point is that those theories of math that give a good explanation of how mathematical statements like "2+2=4" connect with the objects they are about fail to account for how we know about them. And those theories that explain how we know math do poorly in accounting for how statements about math can be true.

One influential philosophy of math is platonism. Platonists hold that numbers are real. They are non-physical entities that exist in a separate realm. Platonism explains well why "There exists one even prime number" is true. It's because quite literally there does exist a prime number that is even, namely, 2, just as the existence of a cat lying on a mat makes it true that "There is a cat on the mat." Problem is that it's really hard to give an explanation of how the human mind can come to know about things like numbers, which (unlike cats), cannot be perceived by our senses. The best that the platonist can do is to posit the existence of some kind of mysterious "intuition" that gives us access to the realm of numbers.

On the other hand, formalists or combinatorialists, as Benacerraf refers to them, reject platonism and argue that mathematical statements are true if they can proved from more basic statements. To be true is simply to be provable using certain rules. Formalism explains how we know that 2+2=4 (because it can be proved from basic statements known as the axioms of Peano arithmetic, which define our concept of natural numbers), but it requires denying that "2+2=4" is true in the same way that propositions like "A cat is on the mat" are true. In essence, if truth is about some correspondence between a statement and the world, there is no such correspondence between mathematical statements and the world for the formalist.

So we are left with a dilemma. Or maybe not. Perhaps the problem with Benacerraf's argument is that he has a deficient theory of knowledge. He assumes that knowledge involves some physical, causal relation between the object of knowledge and the knower. But is that true? That's one question among many that we can discuss on Monday.